Let us consider the problem \((\star )\) . It is proved in [1]} that the same geometric property as in the previous section also holds for regular points if the sequence (REF ) is non-collapsed whose definition is to satisfy \(\mathfrak {m}=a \mathcal {H}^n\)  for some \(a \in (0, \infty )\) , where \(\mathcal {H}^n\)  is the \(n\) -dimensional Hausdorff measure. That is, if \(\mathfrak {m}=a \mathcal {H}^n\) , then any \(n\) -dimensional regular point \(p\)  of \(X\)  has a neighbourhood \(U_p\)  of \(p\)  which is homeomorphic to \(\mathbb {R}^n\) .We say that a point \(p\)  is \(n\) -dimensional regular if any tangent cone at \(p\)  is isometric to \(\big (\mathbb {R}^n, \mathsf {d}_{\mathbb {R}^n}, 0_n\big )\) . In general tangent cones at a point are not unique even in the non-collapsed setting. More strongly, there exists a 5-dimensional non-collapsed Ricci limit space with a base point \(p\)  such that there exist two tangent cones at \(p\)  which are not homeomorphic to each other. See [2]}. Thus it is hard to find nice topological results around singular points, which is very different from the case of sectional curvature (REF ). See also [3]}. This gives a geometric answer to the problem \((\star )\) . See also [4]} for a recent development along this direction.
